Abstract
Numerical modeling of non-Newtonian flows typically involves the coupling between equations of motion characterized by an elliptic character, and the fluid constitutive equation, which is an advection equation linked to the fluid history. Thus, the numerical modeling of short fiber suspensions flows requires a description of the microstructural evolution (fiber orientation) which affects the flow kinematics and that is itself governed by this kinematics. There are different ways to describe the microstructural state. The use of orientation tensors, whose evolution is governed by advection equations, is widely considered. Other possibility is to describe the fiber orientation state from its probability density whose evolution is described by the Fokker–Planck equation (which does not involve any closure relation). The numerical treatment of advection problems is not a simple matter. Moreover, many industrial flows show often steady recirculating areas which introduce some additional difficulties, because in these flows neither boundary conditions nor initial conditions are known. In some of our former papers, we have proposed accurate techniques to solve the linear and non-linear advection equations governing the evolution of the second-order orientation tensor, in steady recirculating flows. These techniques combine a standard treatment of the non-linearity with a more original treatment of the associated linear problems imposing the periodicity condition. In this paper, we generalize those numerical strategies to compute the steady solution of the Fokker–Planck equation (which governs the evolution of the fiber orientation probability distribution) in steady recirculating flows.
Highlights
Numerical modeling of non-Newtonian fluid flows usually involves the coupling between motion equations, which leads to an elliptic problem, and the fluid constitutive equation, which introduces an advection problem related to the fluid history
In [22] we have proved that some linear steady advection problems defined in steady recirculating flows have only one solution when the kinematics differs from a rigid motion
We have proposed firstly an accurate numerical strategy to compute steady solutions of the advection dominated Fokker–Planck equation in steady recirculating flows
Summary
In short fiber suspension flow models, the extra-stress tensor τ depends on the fiber orientation [1,2,3,4]. If Ψ(p) = δ(p − p ), with δ() the Dirac distribution, all the orientation probability is concentrated in the direction definedby p. In this case we can verify that the fourth order orientation tensor can be expressed from the tensorial product of the second-order orientation tensor a a = a ⊗ a,. In the case of a concentrated orientation distribution, the extra-stress tensor can be expressed as τ ̄. When the orientation probability is not concentrated in the same direction at each point of the flow domain, the expression (7) is not exact, and in this case it constitutes a quadratic closure relation
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